Complemented uniform lattices

It is proved that the completion of a complemented modular lattice with respect to a Hausdorff lattice uniformity which is metrizable or exhaustive is a complemented modular lattice. It is then shown that a complete complemented modular lattice endowed with a Hausdorff order continuous lattice uniformity is isomorphic to the product of an arcwise connected complemented lattice and of geometric lattices of finite length each of which endowed with the discrete uniformity. These two results are used to prove a decomposition theorem for modular functions on complemented lattices.